{"paper":{"title":"On conditional expectations in L^p(mu;L^q(nu;X))","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Jan van Neerven, Qi Lu","submitted_at":"2016-06-08T23:12:57Z","abstract_excerpt":"Let $(A,\\mathscr{A},\\mu)$ and $(B,\\mathscr{B},\\nu)$ be probability spaces, let $\\mathscr{F}$ be a sub-$\\sigma$-algebra of the product $\\sigma$-algebra $\\mathscr{A}\\times\\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \\infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\\mathscr{F}$ defines a bounded linear operator from $L^p(\\mu;L^q(\\nu;X))$ onto $L^p_{\\mathscr{F}}(\\mu;L^q(\\nu;X))$, the closed subspace in $L^p(\\mu;L^q(\\nu;X))$ of all functions having a strongly $\\mathscr{F}$-measurable representative."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02780","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}